3.33 \(\int e^{-t} \cos (3 t) \, dt\)

Optimal. Leaf size=27 \[ \frac{3}{10} e^{-t} \sin (3 t)-\frac{1}{10} e^{-t} \cos (3 t) \]

[Out]

-Cos[3*t]/(10*E^t) + (3*Sin[3*t])/(10*E^t)

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Rubi [A]  time = 0.0111672, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {4433} \[ \frac{3}{10} e^{-t} \sin (3 t)-\frac{1}{10} e^{-t} \cos (3 t) \]

Antiderivative was successfully verified.

[In]

Int[Cos[3*t]/E^t,t]

[Out]

-Cos[3*t]/(10*E^t) + (3*Sin[3*t])/(10*E^t)

Rule 4433

Int[Cos[(d_.) + (e_.)*(x_)]*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[(b*c*Log[F]*F^(c*(a + b*x))*C
os[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x] + Simp[(e*F^(c*(a + b*x))*Sin[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x]
/; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 + b^2*c^2*Log[F]^2, 0]

Rubi steps

\begin{align*} \int e^{-t} \cos (3 t) \, dt &=-\frac{1}{10} e^{-t} \cos (3 t)+\frac{3}{10} e^{-t} \sin (3 t)\\ \end{align*}

Mathematica [A]  time = 0.0269654, size = 20, normalized size = 0.74 \[ -\frac{1}{10} e^{-t} (\cos (3 t)-3 \sin (3 t)) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[3*t]/E^t,t]

[Out]

-(Cos[3*t] - 3*Sin[3*t])/(10*E^t)

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Maple [A]  time = 0.007, size = 22, normalized size = 0.8 \begin{align*} -{\frac{{{\rm e}^{-t}}\cos \left ( 3\,t \right ) }{10}}+{\frac{3\,{{\rm e}^{-t}}\sin \left ( 3\,t \right ) }{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(3*t)/exp(t),t)

[Out]

-1/10*exp(-t)*cos(3*t)+3/10*exp(-t)*sin(3*t)

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Maxima [A]  time = 0.936622, size = 23, normalized size = 0.85 \begin{align*} -\frac{1}{10} \,{\left (\cos \left (3 \, t\right ) - 3 \, \sin \left (3 \, t\right )\right )} e^{\left (-t\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3*t)/exp(t),t, algorithm="maxima")

[Out]

-1/10*(cos(3*t) - 3*sin(3*t))*e^(-t)

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Fricas [A]  time = 1.87896, size = 62, normalized size = 2.3 \begin{align*} -\frac{1}{10} \, \cos \left (3 \, t\right ) e^{\left (-t\right )} + \frac{3}{10} \, e^{\left (-t\right )} \sin \left (3 \, t\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3*t)/exp(t),t, algorithm="fricas")

[Out]

-1/10*cos(3*t)*e^(-t) + 3/10*e^(-t)*sin(3*t)

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Sympy [A]  time = 0.458049, size = 20, normalized size = 0.74 \begin{align*} \frac{3 e^{- t} \sin{\left (3 t \right )}}{10} - \frac{e^{- t} \cos{\left (3 t \right )}}{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3*t)/exp(t),t)

[Out]

3*exp(-t)*sin(3*t)/10 - exp(-t)*cos(3*t)/10

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Giac [A]  time = 1.05177, size = 23, normalized size = 0.85 \begin{align*} -\frac{1}{10} \,{\left (\cos \left (3 \, t\right ) - 3 \, \sin \left (3 \, t\right )\right )} e^{\left (-t\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3*t)/exp(t),t, algorithm="giac")

[Out]

-1/10*(cos(3*t) - 3*sin(3*t))*e^(-t)